Bringing memory to Boolean networks: a unifying framework

Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework. We propose novel update modes, the history-based and trapping modes, and provide a comprehensive comparison between them. Furthermore, we show that trapping dynamics, which further generalize the most permissive mode, correspond to a rich class of networks related to transitive dynamics and encompassing commutative networks. Finally, we provide a thorough characterization of the structure of minimal and principal trapspaces, bringing a combinatorial and algebraic understanding of these objects.